Types of solutions and multiplicity results for second order nonlinear boundary value problems
Inara Yermachenko - Daugavpils University, Parades str. 1, LV-5400 Daugavpils, Latvia (email) Abstract: We study the nonlinear BVP $x'' = f(t, x, x')$, (i) $x(0)cos\alpha - x'(0)sin\alpha = 0$, $x(1)cos\Beta - x'(1)sin\Beta = 0$, (ii) provided that $f$ : [0,1] $\times R^2 \rightarrow R$ is continuous together with the partial derivatives $f_(x'), 0 <= \alpha < \pi, 0 < \Beta <= \pi.$ If a quasi-linear ($F$ is bounded) equation $(L_2x)(t) := d/(dt) (e^(2mt)x') + e^(2mt)k^2x = F (t,x,x')$ (iii) can be constructed so that any solution of the problem (iii), (ii) solves also the BVP (i), (ii), then we say that the problem (i), (ii) allows for ($L_2x$)-quasilinearization. We show that if the problem (i), (ii) allows for quasilinearization with respect to essentially different linear parts then the problem (i), (ii) has multiple solutions.
Keywords: Multiple solutions, quasi-linear problems, nonlinear boundary value problems.
Received: September 2006; Revised: June 2007; Published: September 2007. |